Remark 3.3 
We call \(\mathcal {S}\)  semigroup model, because \(\mathcal {S}\)  is a semigroup with respect to \(\circ \)  and \(+\) .
The correct description of \(\mathcal {S}\)  in the sense of universal algebra is an algebra of type \((2,2,0)\)  [1]}.
However, to avoid confusion with the more commonly used notion of an algebra over a ring in commutative algebra (e.g. matrix algebras over fields), we avoid this terminology.
When referring to the actual semigroups \((\mathcal {S},\circ )\)  and \((\mathcal {S},+)\) , we will use the term semigroup instead of semigroup model.

